UPSC CDS Math Volume and Surface Area Questions

Let r and h be the radius and height of the small cone, respectively.
Suppose the radius of the original cone is R.
Now αOAB ~ ΔOCD
∴

OA

OC

=

AB

CD

⇒

h

60

=

r

R

....(1)
Volume of the smaller cone =

1

64

×Volume of the original cone
∴

1 3 πr2h

1 3 πR2×60

=

1

64

⇒(

r

R

)2×

h

60

=

1

64

⇒(

h

60

)3=(

1

4

)3 [Using (1)
⇒h=15cm
Height from the base at which the section is made = 60 – 15 = 45 cm